Lord Williams QC, chairman of the Bar Council and a deputy judge, reserved judgment after Mr Justice Popplewell brought the action over an article in the newspaper Today. He is expected to give his decision next week.
It will be the first time libel damages have been set by a judge rather than a jury and could pave the way for further settlements through arbitration, lawyers believe.
Solicitors say that jury awards are little more than a lottery, with the level of damages fluctuating wildly for no apparent reason. Damages set through arbitration should be more consistent.
Today's report, suggesting that Mr Justice Popplewell 'appeared to nod off' during a murder trial, was published seven weeks ago. The newspaper had apologised for the story, accepting that there was 'no truth' in it, and had offered to make amends. The only issue to be resolved at the arbitration hearing, which was in chambers, is the level of damages. The decision is binding; Today cannot appeal.
The action might set a precedent leading to wider use of arbitration schemes when the defendant admits libel. Last summer, a committee chaired by Lord Justice Neill proposed that a formal arbitration procedure should be established and financed by the media.